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Theorem dilsetN 39658
Description: The set of dilations for a fiducial atom 𝐷. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
dilset.a 𝐴 = (Atomsβ€˜πΎ)
dilset.s 𝑆 = (PSubSpβ€˜πΎ)
dilset.w π‘Š = (WAtomsβ€˜πΎ)
dilset.m 𝑀 = (PAutβ€˜πΎ)
dilset.l 𝐿 = (Dilβ€˜πΎ)
Assertion
Ref Expression
dilsetN ((𝐾 ∈ 𝐡 ∧ 𝐷 ∈ 𝐴) β†’ (πΏβ€˜π·) = {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (π‘“β€˜π‘₯) = π‘₯)})
Distinct variable groups:   π‘₯,𝑓,𝐾   𝑓,𝑀   π‘₯,𝑆   𝐷,𝑓,π‘₯
Allowed substitution hints:   𝐴(π‘₯,𝑓)   𝐡(π‘₯,𝑓)   𝑆(𝑓)   𝐿(π‘₯,𝑓)   𝑀(π‘₯)   π‘Š(π‘₯,𝑓)

Proof of Theorem dilsetN
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 dilset.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
2 dilset.s . . . 4 𝑆 = (PSubSpβ€˜πΎ)
3 dilset.w . . . 4 π‘Š = (WAtomsβ€˜πΎ)
4 dilset.m . . . 4 𝑀 = (PAutβ€˜πΎ)
5 dilset.l . . . 4 𝐿 = (Dilβ€˜πΎ)
61, 2, 3, 4, 5dilfsetN 39657 . . 3 (𝐾 ∈ 𝐡 β†’ 𝐿 = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯)}))
76fveq1d 6904 . 2 (𝐾 ∈ 𝐡 β†’ (πΏβ€˜π·) = ((𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯)})β€˜π·))
8 fveq2 6902 . . . . . . 7 (𝑑 = 𝐷 β†’ (π‘Šβ€˜π‘‘) = (π‘Šβ€˜π·))
98sseq2d 4014 . . . . . 6 (𝑑 = 𝐷 β†’ (π‘₯ βŠ† (π‘Šβ€˜π‘‘) ↔ π‘₯ βŠ† (π‘Šβ€˜π·)))
109imbi1d 340 . . . . 5 (𝑑 = 𝐷 β†’ ((π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯) ↔ (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (π‘“β€˜π‘₯) = π‘₯)))
1110ralbidv 3175 . . . 4 (𝑑 = 𝐷 β†’ (βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯) ↔ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (π‘“β€˜π‘₯) = π‘₯)))
1211rabbidv 3438 . . 3 (𝑑 = 𝐷 β†’ {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯)} = {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (π‘“β€˜π‘₯) = π‘₯)})
13 eqid 2728 . . 3 (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯)}) = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯)})
144fvexi 6916 . . . 4 𝑀 ∈ V
1514rabex 5338 . . 3 {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (π‘“β€˜π‘₯) = π‘₯)} ∈ V
1612, 13, 15fvmpt 7010 . 2 (𝐷 ∈ 𝐴 β†’ ((𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯)})β€˜π·) = {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (π‘“β€˜π‘₯) = π‘₯)})
177, 16sylan9eq 2788 1 ((𝐾 ∈ 𝐡 ∧ 𝐷 ∈ 𝐴) β†’ (πΏβ€˜π·) = {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (π‘“β€˜π‘₯) = π‘₯)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  {crab 3430   βŠ† wss 3949   ↦ cmpt 5235  β€˜cfv 6553  Atomscatm 38767  PSubSpcpsubsp 39001  WAtomscwpointsN 39491  PAutcpautN 39492  DilcdilN 39607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-dilN 39611
This theorem is referenced by:  isdilN  39659
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